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Sensory Adaptation within a Bayesian
Framework for Perception
Alan A. Stocker ∗^ and Eero P. Simoncelli Howard Hughes Medical Institute and Center for Neural Science New York University
Presented at: NIPS-05, Vancouver BC Canada, Dec 2005. Published in: Advances in Neural Information Processing Systems eds. Y. Weiss, B. Sch¨olkopf, and J. Platt vol 18, pp 1291--1298, May 2006 MIT Press, Cambridge MA.
Abstract
We extend a previously developed Bayesian framework for perception to account for sensory adaptation. We first note that the perceptual ef- fects of adaptation seems inconsistent with an adjustment of the inter- nally represented prior distribution. Instead, we postulate that adaptation increases the signal-to-noise ratio of the measurements by adapting the operational range of the measurement stage to the input range. We show that this changes the likelihood function in such a way that the Bayesian estimator model can account for reported perceptual behavior. In particu- lar, we compare the model’s predictions to human motion discrimination data and demonstrate that the model accounts for the commonly observed perceptual adaptation effects of repulsion and enhanced discriminability.
1 Motivation
A growing number of studies support the notion that humans are nearly optimal when per- forming perceptual estimation tasks that require the combination of sensory observations with a priori knowledge. The Bayesian formulation of these problems defines the optimal strategy, and provides a principled yet simple computational framework for perception that can account for a large number of known perceptual effects and illusions, as demonstrated in sensorimotor learning [1], cue combination [2], or visual motion perception [3], just to name a few of the many examples.
Adaptation is a fundamental phenomenon in sensory perception that seems to occur at all processing levels and modalities. A variety of computational principles have been sug- gested as explanations for adaptation. Many of these are based on the concept of maximiz- ing the sensory information an observer can obtain about a stimulus despite limited sensory resources [4, 5, 6]. More mechanistically, adaptation can be interpreted as the attempt of the sensory system to adjusts its (limited) dynamic range such that it is maximally infor- mative with respect to the statistics of the stimulus. A typical example is observed in the retina, which manages to encode light intensities that vary over nine orders of magnitude using ganglion cells whose dynamic range covers only two orders of magnitude. This is achieved by adapting to the local mean as well as higher order statistics of the visual input over short time-scales [7]. ∗corresponding author.
If a Bayesian framework is to provide a valid computational explanation of perceptual processes, then it needs to account for the behavior of a perceptual system, regardless of its adaptation state. In general, adaptation in a sensory estimation task seems to have two fundamental effects on subsequent perception:
- Repulsion: The estimate of parameters of subsequent stimuli are repelled by those of the adaptor stimulus, i.e. the perceived values for the stimulus variable that is subject to the estimation task are more distant from the adaptor value after adaptation. This repulsive effect has been reported for perception of visual speed ( e.g. [8, 9]), direction-of-motion [10], and orientation [11].
- Increased sensitivity: Adaptation increases the observer’s discrimination ability around the adaptor ( e.g. for visual speed [12, 13]), however it also seems to de- crease it further away from the adaptor as shown in the case of direction-of-motion discrimination [14].
In this paper, we show that these two perceptual effects can be explained within a Bayesian estimation framework of perception. Note that our description is at an abstract functional level - we do not attempt to provide a computational model for the underlying mechanisms responsible for adaptation, and this clearly separates this paper from other work which might seem at first glance similar [e.g., 15].
2 Adaptive Bayesian estimator framework
Suppose that an observer wants to estimate a property of a stimulus denoted by the variable θ, based on a measurement m. In general, the measurement can be vector-valued, and is corrupted by both internal and external noise. Hence, combining the noisy information gained by the measurement m with a priori knowledge about θ is advantageous. According to Bayes’ rule
p(θ|m) =
α
p(m|θ)p(θ). (1)
That is, the probability of stimulus value θ given m ( posterior ) is the product of the likeli- hood p(m|θ) of the particular measurement and the prior p(θ). The normalization constant α serves to ensure that the posterior is a proper probability distribution. Under the assump-
tion of a squared-error loss function, the optimal estimate θˆ(m) is the mean of the posterior, thus
θ^ ˆ(m) =
0
θ p(θ|m) dθ. (2)
Note that θˆ(m) describes an estimate for a single measurement m. As discussed in [16], the measurement will vary stochastically over the course of many exposures to the same stimulus, and thus the estimator will also vary. We return to this issue in Section 3.2.
Figure 1a illustrates a Bayesian estimator, in which the shape of the (arbitrary) prior dis- tribution leads on average to a shift of the estimate toward a lower value of θ than the true stimulus value θstim. The likelihood and the prior are the fundamental constituents of the Bayesian estimator model. Our goal is to describe how adaptation alters these constituents so as to account for the perceptual effects of repulsion and increased sensitivity.
Adaptation does not change the prior ...
An intuitively sensible hypothesis is that adaptation changes the prior distribution. Since the prior is meant to reflect the knowledge the observer has about the distribution of occur- rences of the variable θ in the world, repeated viewing of stimuli with the same parameter
m 1
m 2
unadapted adapted
conditionals
likelihoods
1 2
1/SNR
p(m| (^2) θ) '^
p(m| (^) θ 1 ) '
p(m| θ 2 )
p(m| θ 1 )
θ θ θ
p(m| (^) adaptθ ) '
θ
θ adapt
m
θ
m
θ
p(m 1 | )
p(m 1 | )'
p(m 2 | )'
p(m 2 | )
θ
θ
θ
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θ
θ θ adapt
θ 1 θ 2
Figure 2: Measurement noise, conditionals and likelihoods. The two-dimensional condi- tional density, p(m|θ), is shown as a grayscale image for both the unadapted and adapted cases. We assume here that adaptation increases the reliability (SNR) of the measurement around the parameter value of the adaptor. This is balanced by a decrease in SNR of the measurement further away from the adaptor. Because the likelihood is a function of θ (hor- izontal slices, shown plotted at right), this results in an asymmetric change in the likelihood that is in agreement with a repulsive effect on the estimate.
a
0
θ adapt
θ
^θ
b
-90 90 180
0
30
60
θ adapt
^θ
[d
eg]
Figure 3: Repulsion: Model predictions vs. human psychophysics. a) Difference in per- ceived direction in the pre- and post-adaptation condition, as predicted by the model. Post- adaptive percepts of motion direction are repelled away from the direction of the adaptor. b) Typical human subject data show a qualitatively similar repulsive effect. Data (and fit) are replotted from [10].
hood function. The two gray-scale images represent the conditional probability densities, p(m|θ), in the unadapted and the adapted state. They are formed by assuming additive noise on the measurement m of constant variance (unadapted) or with a variance that decreases symmetrically in the vicinity of the adaptor parameter value θadapt, and grows slightly in the region beyond. In the unadapted state, the likelihood is convolutional and the shape and variance are equivalent to the distribution of measurement noise. However, in the adapted state, because the likelihood is a function of θ (horizontal slice through the conditional surface) it is no longer convolutional around the adaptor. As a result, the mean is pushed away from the adaptor, as illustrated in the two graphs on the right. Assuming that the prior distribution is fairly smooth, this repulsion effect is transferred to the posterior distribution, and thus to the estimate.
3 Simulation Results
We have qualitatively demonstrated that an increase in the measurement reliability around the adaptor is consistent with the repulsive effects commonly seen as a result of percep- tual adaptation. In this section, we simulate an adapted Bayesian observer by assuming a simple model for the changes in signal-to-noise ratio due to adaptation. We address both repulsion and changes in discrimination threshold. In particular, we compare our model predictions with previously published data from psychophysical experiments examining human perception of motion direction.
3.1 Repulsion
In the unadapted state, we assume the measurement noise to be additive and normally distributed, and constant over the whole measurement space. Thus, assuming that m and θ live in the same space, the likelihood is a Gaussian of constant width. In the adapted state, we assume a simple functional description for the variance of the measurement noise around the adapter. Specifically, we use a constant plus a difference of two Gaussians,
var〈θˆ|θstim〉, we can now predict how discrimination thresholds should change after adap-
tation. Figure 4a shows the predicted change in discrimination thresholds relative to the un- adapted condition for the same model parameters as in the repulsion example (Figure 3a). Thresholds are slightly reduced at the adaptor, but increase symmetrically for directions further away from the adaptor. For comparison, Figure 4b shows the relative change in dis- crimination thresholds for a typical human subject [14]. Again, the behavior of the human observer is qualitatively well predicted.
4 Discussion
We have shown that adaptation can be incorporated into a Bayesian estimation framework for human sensory perception. Adaptation seems unlikely to manifest itself as a change in the internal representation of prior distributions, as this would lead to perceptual bias effects that are opposite to those observed in human subjects. Instead, we argue that adap- tation leads to an increase in reliability of the measurement in the vicinity of the adapting stimulus parameter. We show that this change in the measurement reliability results in changes of the likelihood function, and that an estimator that utilizes this likelihood func- tion will exhibit the commonly-observed adaptation effects of repulsion and changes in discrimination threshold. We further confirm our model by making quantitative predictions and comparing them with known psychophysical data in the case of human perception of motion direction.
Many open questions remain. The results demonstrated here indicate that a resource alloca- tion explanation is consistent with the functional effects of adaptation, but it seems unlikely that theory alone can lead to a unique quantitative prediction of the detailed form of these effects. Specifically, the constraints imposed by biological implementation are likely to play a role in determining the changes in measurement noise as a function of adaptor pa- rameter value, and it will be important to characterize and interpret neural response changes in the context of our framework. Also, although we have argued that changes in the prior seem inconsistent with adaptation effects, it may be that such changes do occur but are offset by the likelihood effect, or occur only on much longer timescales.
Last, if one considers sensory perception as the result of a cascade of successive processing stages (with both feedforward and feedback connections), it becomes necessary to expand the Bayesian description to describe this cascade [e.g., 18, 19]. For example, it may be possible to interpret this cascade as a sequence of Bayesian estimators, in which the mea- surement of each stage consists of the estimate computed at the previous stage. Adaptation could potentially occur in each of these processing stages, and it is of fundamental interest to understand how such a cascade can perform useful stable computations despite the fact that each of its elements is constantly readjusting its response properties.
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